# Numerical Analysis of Boundary Layer Flow Adjacent to a Thin Needle in Nanofluid with the Presence of Heat Source and Chemical Reaction

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Governing Formula and Modeling

## 3. Stability Analysis

## 4. Graphical Results and Discussion

## 5. Final Remarks

- The heat generation parameter reduces the local heat flux as well as the rate of heat transfer.
- The presence of a chemical reaction increases the rate of mass transfer on the needle surface.
- The Brownian motion parameter diminishes the rate of heat and mass transfers from the needle surface to the flow.
- An increase in the thermophoresis parameter results in an increase in the mass transfer rate, while the reverse effect is noted for the heat transfer rate.
- An increment in the needle thickness leads to decrease the magnitudes of the surface shear stress, local heat flux and local mass flux.
- The dual solutions are likely to exist when the needle surface moves against the free-stream direction, $\epsilon <0$.
- The upper branch solution exhibits stable flow (or solution) and lower branch solution exhibits unstable flow.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

c | Needle size |

C | Fluid concentration (kg m${}^{-3}$) |

${C}_{f}$ | Skin friction coefficient |

${C}_{\infty}$ | Ambient nanoparticle volume fraction |

${C}_{w}$ | Surface volume fraction |

${C}_{p}$ | Specific heat at constant pressure |

${D}_{B}$ | Brownian diffusion coefficient (m${}^{2}$ s${}^{-1}$) |

${D}_{T}$ | Thermophoretic diffusion coefficient (m${}^{2}$ s${}^{-1}$) |

f | Similarity function for velocity |

K | Chemical reaction parameter |

${K}_{0}$ | Chemical reaction coefficient |

${K}^{*}$ | Dimensionless reaction rate |

$Le$ | Lewis number |

$Nb$ | Brownian motion parameter |

$Nt$ | Thermophoresis parameter |

$N{u}_{x}$ | Local Nusselt number |

$Pr$ | Prandtl number |

Q | Heat generation parameter |

${Q}_{0}$ | Heat generation coefficient |

${Q}^{*}$ | Dimensionless heat generation |

r | Cartesian coordinate |

$R{e}_{x}$ | Local Reynolds number |

$S{h}_{x}$ | Local Sherwood number |

T | Fluid temperature (K) |

${T}_{w}$ | Wall temperature (K) |

${T}_{\infty}$ | Ambient temperature (K) |

U | Composite velocity (ms${}^{-1}$) |

${U}_{w}$ | Wall velocity (ms${}^{-1}$) |

${U}_{\infty}$ | Ambient velocity (ms${}^{-1}$) |

u | Velocity in x direction (ms${}^{-1}$) |

v | Velocity in r direction (ms${}^{-1}$) |

x | Cartesian coordinate |

$\alpha $ | Thermal diffusivity (m${}^{2}$ s${}^{-1}$) |

$\eta $ | Similarity independent variable |

$\theta $ | Dimensionless temperature |

$\epsilon $ | Velocity ratio parameter |

$\kappa $ | Ratio of effective heat capacity of nanofluid |

$\rho {C}_{p}$ | Volumetric heat capacity (J K${}^{-1}$) |

$\nu $ | Kinematic viscosity (m${}^{2}$ s${}^{-1}$) |

$\mu $ | Dynamic viscosity (kg m${}^{-1}$s${}^{-1}$) |

$\rho $ | Fluid density (kg m${}^{-3}$) |

$\varphi $ | Dimensionless solid volume fraction |

w | Condition at the wall |

∞ | Ambient condition |

${}^{\prime}$ | Differentiative with respect to $\eta $ |

## References

- Choi, S.U.S. Enhancing thermal conductivity of fluids with nanoparticles. Am. Soc. Mech. Eng. Fluids Eng. Div.
**1995**, 231, 99–105. [Google Scholar] - Wong, K.V.; Leon, O.D. Applications of Nanofluids: Current and Future. Adv. Mech. Eng.
**2010**, 2010, 519659. [Google Scholar] [CrossRef] - Saidur, R.; Leong, K.Y.; Mohammad, H.A. A review on applications and challenges of nanofluids. Renew. Sustain. Energy Rev.
**2011**, 15, 1646–1668. [Google Scholar] [CrossRef] - Huminic, G.; Huminic, A. Application of nanofluids in heat exchangers: A review. Renew. Sustain. Energy Rev.
**2012**, 16, 5625–5638. [Google Scholar] [CrossRef] - Colangelo, G.; Favale, E.; Milanese, M.; Risi, A.D.; Laforgia, D. Cooling of electronic devices: Nanofluids contribution. Appl. Ther. Eng.
**2017**, 127, 421–435. [Google Scholar] [CrossRef] - Buongiorno, J. Convective transport in nanofluids. J. Heat Trans.
**2006**, 128, 240–250. [Google Scholar] [CrossRef] - Tiwari, R.K.; Das, M.K. Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids. Int. J. Heat Mass Trans.
**2007**, 50, 2002–2018. [Google Scholar] [CrossRef] - Khan, W.A.; Pop, I. Boundary-layer flow of a nanofluid past a stretching sheet. Int. J. Heat Mass Trans.
**2010**, 53, 2477–2483. [Google Scholar] [CrossRef] - Makinde, O.D.; Aziz, A. Boundary layer flow of a nanofluid past a stretching sheet with a convective boundary condition. Int. J. Ther. Sci.
**2011**, 50, 1326–1332. [Google Scholar] [CrossRef] - Bachok, N.; Ishak, A.; Pop, I. Unsteady boundary-layer flow and heat transfer of a nanofluid over a permeable stretching/shrinking sheet. Int. J. Heat Mass Trans.
**2012**, 55, 2102–2109. [Google Scholar] [CrossRef] - Das, K.; Duari, P.R.; Kundu, P.K. Nanofluid flow over an unsteady stretching surface in presence of thermal radiation. Alex. Eng. J.
**2014**, 53, 737–745. [Google Scholar] [CrossRef] [Green Version] - Mabood, F.; Khan, W.A.; Ismail, A.I.M. MHD boundary layer flow and heat transfer of nanofluids over a nonlinear stretching sheet: A numerical study. J. Magn. Magn. Mater.
**2015**, 374, 569–576. [Google Scholar] [CrossRef] - Naramgari, S.; Sulochana, C. MHD flow over a permeable stretching/shrinking sheet of a nanofluid with suction/injection. Alex. Eng. J.
**2016**, 55, 819–827. [Google Scholar] [CrossRef] [Green Version] - Pandey, A.K.; Kumar, M. Boundary layer flow and heat transfer analysis on Cu-water nanofluid flow over a stretching cylinder with slip. Alex. Eng. J.
**2017**, 56, 671–677. [Google Scholar] [CrossRef] - Mustafa, M. MHD nanofluid flow over a rotating disk with partial slip effects: Buongiorno model. Int. J. Heat Mass Trans.
**2017**, 108, 1910–1916. [Google Scholar] [CrossRef] - Jyothi, K.; Reddy, P.S.; Reddy, M.S. Influence of magnetic field and thermal radiation on convective flow of SWCNTs-water and MWCNTs-water nanofluid between rotating stretchable disks with convective boundary conditions. Adv. Powder Technol.
**2018**, 331, 326–337. [Google Scholar] [CrossRef] - Bakar, N.A.A.; Bachok, N.; Arifin, N.M.; Pop, I. Stability analysis on the flow and heat transfer of nanofluid past a stretching/shrinking cylinder with suction effect. Results Phys.
**2018**, 9, 1335–1344. [Google Scholar] [CrossRef] - Griffith, R.M. Velocity, temperature and concentration distributions during fiber spinning. Ind. Eng. Chem. Fundam.
**1964**, 3, 245–250. [Google Scholar] [CrossRef] - Chin, D.T. Mass transfer to a continuousmoving sheet electrode. J. Electrochem. Soc.
**1975**, 122, 643–646. [Google Scholar] [CrossRef] - Gorla, R.S.R. Unsteady mass transfer in the boundary layer on a continuous moving sheet electrode. J. Electrochem. Soc.
**1978**, 125, 865–869. [Google Scholar] [CrossRef] - Damseh, R.A.; Al-Odat, M.Q.; Chamkha, A.J.; Shannak, B.A. Combined effect of heat generation or absorption and first-order chemical reaction on micropolar fluid flows over a uniformly stretched permeable surface. Int. J. Therm. Sci.
**2009**, 48, 1658–1663. [Google Scholar] [CrossRef] - Magyari, E.; Chamkha, A.J. Combined effect of heat generation or absorption and first-order chemical reaction on micropolar fluid flows over a uniformly stretched permeable surface: The full analytical solution. Int. J. Therm. Sci.
**2010**, 49, 1821–1828. [Google Scholar] [CrossRef] - Mabood, F.; Shateyi, S.; Rashidi, M.M.; Momoniat, E.; Freidoonimehr, N. MHD stagnation point flow heat and mass transfer of nanofluids in porous medium with radiation, viscous dissipation and chemical reaction. Adv. Powder Technol.
**2016**, 27, 742–749. [Google Scholar] [CrossRef] - Eid, M.R. Chemical reaction effect on MHD boundary-layer flow of two-phase nanofluid model over an exponentially stretching sheet with a heat generation. J. Mol. Liq.
**2016**, 220, 718–725. [Google Scholar] [CrossRef] - Ibrahim, S.M.; Lorenzini, G.; Vijaya Kumar, P.; Raju, C.S.K. Influence of chemical reaction and heat source on dissipative MHD mixed convection flow of a Casson nanofluid over a nonlinear permeable stretching sheet. Int. J. Heat Mass Trans.
**2017**, 111, 346355. [Google Scholar] [CrossRef] - Nayak, M.K.; Akbar, N.S.; Tripathi, D.; Khan, Z.H.; Pandey, V.S. MHD 3D free convective flow of nanofluid over an exponentially stretching sheet with chemical reaction. Adv. Powder Technol.
**2017**, 28, 2159–2166. [Google Scholar] [CrossRef] - Sithole, H.; Mondal, H.; Goqo, S.; Sibanda, P.; Motsa, S. Numerical simulation of couple stress nanofluid flow in magneto-porous medium with thermal radiation and a chemical reaction. Appl. Math. Comput.
**2018**, 339, 820–836. [Google Scholar] [CrossRef] - Khan, M.; Shahid, A.; Malik, M.Y.; Salahuddin, T. Chemical reaction for Carreau-Yasuda nanofluid flow past a nonlinear stretching sheet considering Joule heating. Results Phys.
**2018**, 8, 1124–1130. [Google Scholar] [CrossRef] - Zeeshan, A.; Shehzad, N.; Ellahi, R. Analysis of activation energy in Couette-Poiseuille flow of nanofluid in the presence of chemical reaction and convective boundary conditions. Results Phys.
**2018**, 8, 502–512. [Google Scholar] [CrossRef] - Hayat, T.; Kiyani, M.Z.; Alsaedi, A.; Ijaz Khan, M.; Ahmad, I. Mixed convective three-dimensional flow of Williamson nanofluid subject to chemical reaction. Int. J. Heat Mass Trans.
**2018**, 127, 422–429. [Google Scholar] [CrossRef] - Lee, L.L. Boundary layer over a thin needle. Phys. Fluids
**1967**, 10, 1820–1822. [Google Scholar] [CrossRef] - Narain, J.P.; Uberoi, S.M. Combined forced and free-convection heat transfer from vertical thin needles in a uniform stream. Phys. Fluids
**1973**, 15, 1879–1882. [Google Scholar] [CrossRef] - Wang, C.Y. Mixed convection on a vertical needle with heated tip. Phys. Fluids
**1990**, 2, 622–625. [Google Scholar] [CrossRef] - Ishak, A.; Nazar, R.; Pop, I. Boundary layer flow over a continuously moving thin needle in a parallel free stream. Chin. Phys. Lett.
**2007**, 24, 2895–2897. [Google Scholar] [CrossRef] - Ahmad, S.; Arifin, N.M.; Nazar, R.; Pop, I. Mixed convection boundary layer flow along vertical thin needles: Assisting and opposing flows. Int. Commun. Heat Mass Trans.
**2008**, 35, 157–162. [Google Scholar] [CrossRef] - Afridi, M.I.; Qasim, M. Entropy generation and heat transfer in boundary layer flow over a thin needle moving in a parallel stream in the presence of nonlinear Rosseland radiation. Int. J. Therm. Sci.
**2018**, 123, 117–128. [Google Scholar] [CrossRef] - Grosan, T.; Pop, I. Forced Convection Boundary Layer Flow Past Nonisothermal Thin Needles in Nanofluids. J. Heat Trans.
**2011**, 133. [Google Scholar] [CrossRef] - Trimbitas, R.; Grosan, T.; Pop, I. Mixed convection boundary layer flow along vertical thin needles in nanofluids. Int. J. Numer. Methods Heat Fluid Flow
**2014**, 24, 579–594. [Google Scholar] [CrossRef] - Hayat, T.; Khan, M.I.; Farooq, M.; Yasmeen, T.; Alsaedi, A. Water-carbon nanofluid flow with variable heat flux by a thin needle. J. Mol. Liq.
**2016**, 224, 786–791. [Google Scholar] [CrossRef] - Soid, S.K.; Ishak, A.; Pop, I. Boundary layer flow past a continuously moving thin needle in a nanofluid. Appl. Therm. Eng.
**2017**, 114, 58–64. [Google Scholar] [CrossRef] - Krishna, P.M.; Sharma, R.P.; Sandeep, N. Boundary layer analysis of persistent moving horizontal needle in Blasius and Sakiadis magnetohydrodynamic radiative nanofluid flows. Nucl. Eng. Technol.
**2017**, 49, 1654–1659. [Google Scholar] [CrossRef] - Ahmad, R.; Mustafa, M.; Hina, S. Buongiorno’s model for fluid flow around a moving thin needle in a flowing nanofluid: A numerical study. Chin. J. Phys.
**2017**, 55, 1264–1274. [Google Scholar] [CrossRef] - Salleh, S.N.A.; Bachok, N.; Arifin, N.M.; Ali, F.M.; Pop, I. Magnetohydrodynamics flow past a moving vertical thin needle in a nanofluid with stability analysis. Energies
**2018**, 11, 3297. [Google Scholar] [CrossRef] - Weidman, P.D.; Kubitschek, D.G.; Davis, A.M.J. The effect of transpiration on self-similar boundary layer flow over moving surfaces. Int. J. Eng. Sci.
**2006**, 44, 730–737. [Google Scholar] [CrossRef] - Rosca, A.V.; Pop, I. Flow and heat transfer over a vertical permeable stretching/shrinking sheet with a second order slip. Int. J. Heat Mass Trans.
**2013**, 60, 355–364. [Google Scholar] [CrossRef] - Harris, S.D.; Ingham, D.B.; Pop, I. Mixed convection boundary-layer flow near the stagnation point on a vertical surface in a porous medium: Brinkman model with slip. Transp. Porous Media
**2009**, 77, 267–285. [Google Scholar] [CrossRef]

**Figure 2.**Sample of (

**a**) velocity, (

**b**) temperature and (

**c**) concentration profiles for several values of needle thickness c.

**Figure 3.**Variation of surface (

**a**) shear stress, (

**b**) local heat flux and (

**c**) local mass flux with velocity ratio parameter $\epsilon $ for several values of needle thickness c.

**Figure 4.**Sample of (

**a**) temperature and (

**b**) concentration profiles for several values of heat generation parameter Q.

**Figure 5.**Variation of surface (

**a**) local heat flux and (

**b**) local mass flux with velocity ratio parameter $\epsilon $ for several values of heat generation parameter Q.

**Figure 8.**Sample of (

**a**) temperature and (

**b**) concentration profiles for several values of Brownian motion parameter $Nb$.

**Figure 9.**Sample of (

**a**) temperature and (

**b**) concentration profiles for several values of thermophoresis parameter $Nt$.

**Table 1.**Comparison values of shear stress ${f}^{\u2033}\left(c\right)$ when $\epsilon =Le=Q=K=0$ for some of the thickness of the needle c when $Pr=1$.

c | Ahmad et al. [42] | Salleh et al. [43] | Current Study |
---|---|---|---|

0.01 | 8.4924360 | 8.4924452 | 8.4924453 |

0.1 | 1.2888171 | 1.2888299 | 1.2888300 |

0.15 | - | - | 0.9383388 |

0.2 | - | - | 0.7515725 |

**Table 2.**Effects of thermphoresis parameter $Nt$ and Brownian motion parameter $Nb$ on the numerical values of local Nusselt number, ${\left(R{e}_{x}\right)}^{-1/2}N{u}_{x}$ for $Q=0.1$ and $Q=0.2$ when $\epsilon =-1.0$, $K=0.2$, $c=0.1$, $Pr=2$ and $Le=1$.

Heat Generation Parameter | Thermphoresis Parameter | ${\left({\mathit{Re}}_{\mathit{x}}\right)}^{-1/2}{\mathit{Nu}}_{\mathit{x}}=-2{\mathit{c}}^{1/2}{\mathit{\theta}}^{\prime}\left(\mathit{c}\right)$ | ||
---|---|---|---|---|

$\mathit{Nb}=\mathbf{0.1}$ | $\mathit{Nb}=\mathbf{0.3}$ | $\mathit{Nb}=\mathbf{0.5}$ | ||

0.1 | 0.1 | 1.208880 | 0.959223 | 0.749362 |

0.3 | 0.989170 | 0.777343 | 0.601318 | |

0.5 | 0.805833 | 0.627349 | 0.480535 | |

0.2 | 0.1 | 1.078738 | 0.832762 | 0.628219 |

0.3 | 0.863788 | 0.656596 | 0.486510 | |

0.5 | 0.685682 | 0.512462 | 0.371923 |

**Table 3.**Effects of thermphoresis parameter $Nt$ and Brownian motion parameter $Nb$ on the numerical values of local Sherwood number, ${\left(R{e}_{x}\right)}^{-1/2}S{h}_{x}$ for $K=0.1$ and $K=0.2$ when $\epsilon =-1.0$, $Q=0.2$, $c=0.1$, $Pr=2$ and $Le=1$.

Chemical Reaction Parameter | Thermphoresis Parameter | ${\left({\mathit{Re}}_{\mathit{x}}\right)}^{-1/2}{\mathit{Sh}}_{\mathit{x}}=-2{\mathit{c}}^{1/2}{\mathit{\varphi}}^{\prime}\left(\mathit{c}\right)$ | ||
---|---|---|---|---|

$\mathit{Nb}=\mathbf{0.1}$ | $\mathit{Nb}=\mathbf{0.3}$ | $\mathit{Nb}=\mathbf{0.5}$ | ||

0.1 | 0.1 | 2.005009 | 1.825444 | 1.781448 |

0.3 | 3.271205 | 2.343083 | 2.135890 | |

0.5 | 5.055702 | 2.990494 | 2.546082 | |

0.2 | 0.1 | 2.085015 | 1.898138 | 1.852426 |

0.3 | 3.362939 | 2.418572 | 2.207592 | |

0.5 | 5.144680 | 3.063527 | 2.615317 |

**Table 4.**Smallest eigenvalues $\gamma $ for several values of chemical reaction parameter K, heat generation parameter Q and velocity ratio parameter $\epsilon $ for $c=0.1$ and $c=0.2$ when $Nb=Nt=0.1$, $Pr=2$ and $Le=1$.

K = Q | c | $\mathit{\epsilon}$ | Upper Branch | Lower Branch |
---|---|---|---|---|

0.1 | 0.1 | −4.1994 | 0.0471 | −0.0449 |

−4.199 | 0.0481 | −0.0458 | ||

−4.19 | 0.0668 | −0.0625 | ||

0.2 | −2.7424 | 0.0150 | −0.0147 | |

−2.742 | 0.0175 | −0.0170 | ||

−2.74 | 0.0265 | −0.0255 | ||

0.2 | 0.1 | −4.1246 | 0.1444 | −0.1254 |

−4.124 | 0.1449 | −0.1258 | ||

−4.12 | 0.1484 | −0.1284 | ||

0.2 | −2.7136 | 0.0793 | −0.0706 | |

−2.713 | 0.0801 | −0.0713 | ||

−2.71 | 0.0841 | −0.0744 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Salleh, S.N.A.; Bachok, N.; Arifin, N.M.; Ali, F.M.
Numerical Analysis of Boundary Layer Flow Adjacent to a Thin Needle in Nanofluid with the Presence of Heat Source and Chemical Reaction. *Symmetry* **2019**, *11*, 543.
https://doi.org/10.3390/sym11040543

**AMA Style**

Salleh SNA, Bachok N, Arifin NM, Ali FM.
Numerical Analysis of Boundary Layer Flow Adjacent to a Thin Needle in Nanofluid with the Presence of Heat Source and Chemical Reaction. *Symmetry*. 2019; 11(4):543.
https://doi.org/10.3390/sym11040543

**Chicago/Turabian Style**

Salleh, Siti Nur Alwani, Norfifah Bachok, Norihan Md Arifin, and Fadzilah Md Ali.
2019. "Numerical Analysis of Boundary Layer Flow Adjacent to a Thin Needle in Nanofluid with the Presence of Heat Source and Chemical Reaction" *Symmetry* 11, no. 4: 543.
https://doi.org/10.3390/sym11040543